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I took the liberty of copying below(Dave's International Job Board) educational job requirements for a TEFL teaching position in Abu Dhabi, UAE for a vocational post secondary (college) institution. It asks for ESL certification along with a Masters degree strongly preferred.

Is the CELTA necessary? Most Middle East EFL employers want their teaching candidates to be TEFL certified, not necessarily with a CELTA though that certification is most internationally recognized.

So, with your research indicating TEFL certification is a must for a teaching position in the UAE, and my random job specs indicating certification is also a must, the conclusion is obvious.

Even with an MA, the OP likely wouldn't qualify for that vocational school position given that preference is for a 'licensed teacher'---one who possesses a government license to teach from their home country/state. (Hence, the reference to content or 'subject area.') Plus, for university positions in this region, employers want to see teaching experience that was gained post-TEFL cert or MA as verification of EFL teacher training. Unfortunately, the OP's been teaching for 4+ years without having received that critical supervised teacher training.